Bevan point

[1] Bevan posed the problem of proving this in 1804, in a mathematical problem column in The Mathematical Repository.

[1][2] The problem was solved in 1806 by John Butterworth.

[2] The Bevan point M of triangle △ABC has the same distance from its Euler line e as its incenter I.

where R denotes the radius of the circumcircle and a, b, c the sides of △ABC.

[2] The Bevan is point is also the midpoint of the line segment NL connecting the Nagel point N and the de Longchamps point L.[1] The radius of the Bevan circle is 2R, that is twice the radius of the circumcircle.

Reference triangle △ ABC
Excentral triangle △ M A M B M C of △ ABC
Circumcircle of △ M A M B M C ( Bevan circle of △ ABC , centered at Bevan point M )
Reference triangle △ ABC
Excentral triangle △ M A M B M C of △ ABC
Bevan circle k M of △ ABC (centered at Bevan point M )
Euler line e , on which circumcenter O , orthocenter H , centroid G , and de Longchamps point L lie
Other points: incenter I , Nagel point N